Velocity of sound in a medium is given by `A^(1//2) d^(-1//2)`, where 'd' is density of medium. Then A represents

Options

A. Elastic constant
B. Energy
C. Volume
D. Compressibility

Answer

To solve the problem, we need to analyze the given equation for the velocity of sound in a medium, which is expressed as:

\[ v = A^{1/2} d^{-1/2} \]

where \( v \) is the velocity of sound and \( d \) is the density of the medium. Our goal is to determine what \( A \) represents.

### Step 1: Rearranging the equation
We start by squaring both sides of the equation to eliminate the square root:

\[ v^2 = A \cdot d^{-1} \]

This can be rewritten as:

\[ v^2 = \frac{A}{d} \]

### Step 2: Isolating \( A \)
Next, we can isolate \( A \) by multiplying both sides by \( d \):

\[ A = v^2 \cdot d \]

### Step 3: Finding the dimensions of \( A \)
Now, we need to express \( A \) in terms of its dimensional formula. We know the dimensions of \( v \) (velocity) and \( d \) (density):

- The dimension of velocity \( v \) is given by:
\[ [v] = L T^{-1} \]

- The dimension of density \( d \) is given by:
\[ [d] = M L^{-3} \]

### Step 4: Calculating the dimensions of \( A \)
Substituting the dimensions of \( v \) and \( d \) into the equation for \( A \):

\[ A = (L T^{-1})^2 \cdot (M L^{-3}) \]

Calculating this gives:

\[ A = L^2 T^{-2} \cdot M L^{-3} \]

Now, we can combine the dimensions:

\[ A = M L^{2 - 3} T^{-2} = M L^{-1} T^{-2} \]

### Step 5: Identifying what \( A \) represents
The dimensional formula \( M L^{-1} T^{-2} \) corresponds to the dimensional formula for the elastic constant (also known as the modulus of elasticity), which is defined as stress (force per unit area) divided by strain (dimensionless).

### Conclusion
Thus, we conclude that \( A \) represents the elastic constant of the medium.